Asymmetric Information Security Games 1/43

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1 Asymmetric Information Security Games Jeff S. Shamma with Lichun Li & Malachi Jones & IPAM Graduate Summer School Games and Contracts for Cyber-Physical Security 7 23 July 2015 Jeff S. Shamma Asymmetric Information Security Games 1/43

2 Asymmetric information games Motivation: One player has superior information Attacker knowns own skill set Defender knows resource characteristics Jeff S. Shamma Asymmetric Information Security Games 2/43

3 Asymmetric information games Motivation: One player has superior information Attacker knowns own skill set Defender knows resource characteristics Tension: Exploiting information also reveals information Jeff S. Shamma Asymmetric Information Security Games 2/43

4 Asymmetric information games Motivation: One player has superior information Attacker knowns own skill set Defender knows resource characteristics Tension: Exploiting information also reveals information Solution: Randomized deception Deliberately do not utilize best resources Jeff S. Shamma Asymmetric Information Security Games 2/43

5 Illustration: Network interdiction System: One high capacity resource and several low capacity (unknown to attacker) Attacker: Observes usage (binary) during Phase I Disables selected resource for Phase II Tension: Initial vs future usage Jeff S. Shamma Asymmetric Information Security Games 3/43

6 Framework: Repeated games Players: Maximizer Row with actions Minimizer Col with actions a A = {1, 2,..., A } b B = {1, 2,..., B } Jeff S. Shamma Asymmetric Information Security Games 4/43

7 Framework: Repeated games Players: Maximizer Row with actions Minimizer Col with actions a A = {1, 2,..., A } b B = {1, 2,..., B } Stages: t = 0, 1, 2,... Jeff S. Shamma Asymmetric Information Security Games 4/43

8 Framework: Repeated games Players: Maximizer Row with actions Minimizer Col with actions a A = {1, 2,..., A } b B = {1, 2,..., B } Stages: t = 0, 1, 2,... History: h t = {(a 0, b 0), (a 1, b 1),..., (a t 1, b t 1)} H t Set of finite histories: H Jeff S. Shamma Asymmetric Information Security Games 4/43

9 Repeated games, cont Stage payoff: Matrix M = [m ab ] m ab = Payoff to Row under action pair (a, b) m ab = Penalty to Col under action pair (a, b) Jeff S. Shamma Asymmetric Information Security Games 5/43

10 Repeated games, cont Stage payoff: Matrix M = [m ab ] m ab = Payoff to Row under action pair (a, b) m ab = Penalty to Col under action pair (a, b) State: k K = {1, 2,..., K } } Index of stage game: {M 1, M 2,..., M K Jeff S. Shamma Asymmetric Information Security Games 5/43

11 Repeated games, cont Stage payoff: Matrix M = [m ab ] m ab = Payoff to Row under action pair (a, b) m ab = Penalty to Col under action pair (a, b) State: k K = {1, 2,..., K } } Index of stage game: {M 1, M 2,..., M K Asymmetric information: } At t = 0, nature selects M {M 1, M 2,..., M K } Prior probabilities {p 1, p 2,..., p K Row informed of selected game Jeff S. Shamma Asymmetric Information Security Games 5/43

12 Repeated games, cont Behavioral strategies: σ : H K (A) (Row) τ : H (B) (Col) Jeff S. Shamma Asymmetric Information Security Games 6/43

13 Repeated games, cont Behavioral strategies: σ : H K (A) (Row) τ : H (B) (Col) Note: Do not measure payoffs! Jeff S. Shamma Asymmetric Information Security Games 6/43

14 Repeated games, cont Behavioral strategies: σ : H K (A) (Row) τ : H (B) (Col) Note: Do not measure payoffs! T -stage game, Γ T (p): (Γ (p) later...) [ ] T 1 1 γ t (σ, τ) = E M k (a t, b t ) T t=0 Jeff S. Shamma Asymmetric Information Security Games 6/43

15 Extensive literature Foundations: Aumann & Maschler (1967), Repeated games with incomplete information: A survey of resent results, Report to US Arms Control and Disarmament Agency. Surveys: Zamir (1992), Repeated games of incomplete information: Zero-sum, Handbook of Game Theory, v. I. Laraki & Sorin (2014), Advances in zero-sum dynamic games, Handbook of Game Theory, v. IV. Monographs: Aumann & Maschler (1967/1995), Repeated Games with Incomplete Information. Mertens, Sorin & Zamir (1994/2015), Repeated Games. Sorin (2002), A First Course on Zero-Sum Repeated Games. Variations: Evolving state, signal monitoring, two-sided incomplete information,... Jeff S. Shamma Asymmetric Information Security Games 7/43

16 Pre-example: Dominant strategies L R T 4 3 B 2 1 Case I L R T 4 2 B 1 3 Case II Case I: T is a dominant strategy, i.e., oblivious best response Case II: No dominant strategy, i.e., contingent best response Jeff S. Shamma Asymmetric Information Security Games 8/43

17 Example I: Non-revelation L R T 1 0 B 0 0 M 1 L R T 0 0 B 0 1 M 2 Should Row use (weakly) dominant strategy? Jeff S. Shamma Asymmetric Information Security Games 9/43

18 Example I: Non-revelation L R T 1 0 B 0 0 M 1 L R T 0 0 B 0 1 M 2 Should Row use (weakly) dominant strategy? Dominant strategy payoff stream: 1-or-0, 0, 0,... Jeff S. Shamma Asymmetric Information Security Games 9/43

19 Example I: Non-revelation L R T 1 0 B 0 0 M 1 L R T 0 0 B 0 1 M 2 Should Row use (weakly) dominant strategy? Dominant strategy payoff stream: 1-or-0, 0, 0,... Compare to perpetual strategy: Col faced with averaged game ( 1/2 0 ) 0 1/2 Expected payoff of 1/4 Jeff S. Shamma Asymmetric Information Security Games 9/43

20 Example I: Non-revelation L R T 1 0 B 0 0 M 1 L R T 0 0 B 0 1 M 2 Should Row use (weakly) dominant strategy? Dominant strategy payoff stream: 1-or-0, 0, 0,... Compare to perpetual strategy: Col faced with averaged game ( 1/2 0 ) 0 1/2 Expected payoff of 1/4 Conclusion: Row better off ignoring information Jeff S. Shamma Asymmetric Information Security Games 9/43

21 Example II: Full revelation L R T -1 0 B 0 0 M 1 L R T 0 0 B 0-1 M 2 Dominant strategy (fully revealing) payoff: 0 Non-revealing game: ( 1/2 ) 0 0 1/2 has expected payoff 1/4 Conclusion: Row better off fully revealing Jeff S. Shamma Asymmetric Information Security Games 10/43

22 Example III: Partial revelation L C R T B M 1 L C R T B M 2 Dominant strategy (long run) payoff: 0 Non-revealing game: ( ) has expected payoff 0 Can Row do better? Jeff S. Shamma Asymmetric Information Security Games 11/43

23 Example III: Partial revelation, cont L C R T B M 1 State dependent lottery (h, t): k = 1 probabilities: (3/4, 1/4) k = 2 probabilities: (1/4, 3/4) Outcome dependent strategy: h: Play T forever t: Play B forever L C R T B M 2 Jeff S. Shamma Asymmetric Information Security Games 12/43

24 Example III: Partial revelation, cont L C R T B M 1 L C R T B After stage 0, Col knows outcome Pr [ k = 1 ] h = 3/4 leading to average game ( 3 1 ) in which Row plays T. Pr [ k = 1 ] t = 1/4 leading to average game ( 3 1 ) in which Row plays B. Long run payoff: 1 M 2 Jeff S. Shamma Asymmetric Information Security Games 13/43

25 Outline Selected basic results Computational approaches Jeff S. Shamma Asymmetric Information Security Games 14/43

26 Review: Finite zero-sum games Payoff matrix: M = [m ab ] Jeff S. Shamma Asymmetric Information Security Games 15/43

27 Review: Finite zero-sum games Payoff matrix: M = [m ab ] Mixed strategies: x, y x(a) = Pr [a] & y(b) = Pr [b] x T My = Expected payoff to Row under strategies (x, y) x T My = Expected penalty to Col under strategies (x, y) Jeff S. Shamma Asymmetric Information Security Games 15/43

28 Review: Finite zero-sum games Payoff matrix: M = [m ab ] Mixed strategies: x, y x(a) = Pr [a] & y(b) = Pr [b] x T My = Expected payoff to Row under strategies (x, y) x T My = Expected penalty to Col under strategies (x, y) Security levels and strategies: v = max x = max l x,l min x T My y x T M l 1 T x v = min y = min y,l l My l 1 y max x T My x Jeff S. Shamma Asymmetric Information Security Games 15/43

29 Review: Finite zero-sum games Payoff matrix: M = [m ab ] Mixed strategies: x, y x(a) = Pr [a] & y(b) = Pr [b] x T My = Expected payoff to Row under strategies (x, y) x T My = Expected penalty to Col under strategies (x, y) Security levels and strategies: v = max x = max l x,l min x T My y x T M l 1 T x v = min y = min y,l l My l 1 y max x T My x Value: val[m] = v = v Jeff S. Shamma Asymmetric Information Security Games 15/43

30 Analysis: Finite horizon Pure strategies: σ : H K A (Row) τ : H B (Col) Jeff S. Shamma Asymmetric Information Security Games 16/43

31 Analysis: Finite horizon Pure strategies: σ : H K A (Row) τ : H B (Col) Strategic form conversion: Enumerate all pure strategies Define M(p) as associated (large) matrix game for p (K) Jeff S. Shamma Asymmetric Information Security Games 16/43

32 Analysis: Finite horizon Pure strategies: σ : H K A (Row) τ : H B (Col) Strategic form conversion: Enumerate all pure strategies Define M(p) as associated (large) matrix game for p (K) Consequences: Value v t(p) = val[m(p)] exists along with associated (mixed) security strategies Equivalence to behavioral strategies (Kuhn s theorem) Jeff S. Shamma Asymmetric Information Security Games 16/43

33 Non-revealing game & u(p) Average game: D(p) = k p k M k & u(p) = val[d(p)] Jeff S. Shamma Asymmetric Information Security Games 17/43

34 Non-revealing game & u(p) Average game: D(p) = k p k M k & u(p) = val[d(p)] Claim: v t (p) u(p) Proof: Row plays security strategy for D(p) Jeff S. Shamma Asymmetric Information Security Games 17/43

35 Non-revealing game & u(p) Average game: D(p) = k p k M k & u(p) = val[d(p)] Claim: v t (p) u(p) Proof: Row plays security strategy for D(p) Jeff S. Shamma Asymmetric Information Security Games 17/43

36 Computing u(p) u(p) = max min x T x y x T ( k max l x,l ) p k M k ( ) p k M k y k l 1 T x Jeff S. Shamma Asymmetric Information Security Games 18/43

37 Cavu(p) Claim: Suppose L p = λ l p l l=1 There exists a Row strategy such that L v t (p) λ l u(p l ) l=1 Jeff S. Shamma Asymmetric Information Security Games 19/43

38 Cavu(p) Claim: Suppose L p = λ l p l l=1 There exists a Row strategy such that L v t (p) λ l u(p l ) l=1 Implication: By optimally selecting mixtures v T (p) Cavu(p) where Cavu(p) u(p) is pointwise smallest concave function Jeff S. Shamma Asymmetric Information Security Games 19/43

39 Cavu(p), cont Jeff S. Shamma Asymmetric Information Security Games 20/43

40 Belief splitting For p, p 1,..., p N, suppose L p = λ l p l l=1 Define joint distribution over {1, 2,..., L} {1, 2,..., K } Q(l, k) = λ l pl k Jeff S. Shamma Asymmetric Information Security Games 21/43

41 Belief splitting For p, p 1,..., p N, suppose L p = λ l p l l=1 Define joint distribution over {1, 2,..., L} {1, 2,..., K } Q(l, k) = λ l pl k Properties: Pr [k] = p k Pr [l] = λ l Pr [ k l ] = p k l Pr [ l k ] λl p k l 1 2 p λ1p1 λ2p2 λlpl K λ 1 2 L Jeff S. Shamma Asymmetric Information Security Games 21/43

42 Proof of claim Starting point: p = L l=1 λ lp l Row strategy: 1 Let x l be Row optimal strategy for u(p l ) 2 Select l λ l p k l (k-dependent lottery) 3 Play selected x l Jeff S. Shamma Asymmetric Information Security Games 22/43

43 Proof of claim Starting point: p = L l=1 λ lp l Row strategy: 1 Let x l be Row optimal strategy for u(p l ) 2 Select l λ l p k l (k-dependent lottery) 3 Play selected x l Assess Col response as if observed l min y p k( Pr [ l k ] ) TM x k l y k l min y l,l=1,2,...,l = l λ l u(p l ) p k( Pr [ l k ] ) TM x k l y l k Jeff S. Shamma Asymmetric Information Security Games 22/43

44 Belief updates for Col Belief splitting: Posterior belief is p l with probability λ l Jeff S. Shamma Asymmetric Information Security Games 23/43

45 Belief updates for Col Belief splitting: Posterior belief is p l with probability λ l General setup: Prior beliefs p over K k-dependent strategy for Row, X = (x 1,..., x K ) (A) K x k a = Pr [ a k ] Jeff S. Shamma Asymmetric Information Security Games 23/43

46 Belief updates for Col Belief splitting: Posterior belief is p l with probability λ l General setup: Prior beliefs p over K k-dependent strategy for Row, X = (x 1,..., x K ) (A) K x k a = Pr [ a k ] Computations: Probability Row plays a: π(a; X, p) = k Pr [ a ] k Pr [k] = xa k p k k Posterior belief after Row plays a: B k (a; X, p) = Pr [ a k ] Pr [k] Pr [a] = x k a p k π(a; X, p) Jeff S. Shamma Asymmetric Information Security Games 23/43

47 Naive strategy for Col Naive defense: Define p k t = Pr [ k h t ] (requires Row strategy σ) Set τ t(h t) to be security strategy for D(p t) Jeff S. Shamma Asymmetric Information Security Games 24/43

48 Naive strategy for Col Naive defense: Define p k t = Pr [ k h t ] (requires Row strategy σ) Set τ t(h t) to be security strategy for D(p t) Claim: E [γ t (σ, τ)] Cavu(p) + M p k (1 p k ) t k Jeff S. Shamma Asymmetric Information Security Games 24/43

49 Naive strategy for Col Naive defense: Define p k t = Pr [ k h t ] (requires Row strategy σ) Set τ t(h t) to be security strategy for D(p t) Claim: E [γ t (σ, τ)] Cavu(p) + M p k (1 p k ) t k Implication: Cavu(p) v t (p) Cavu(p) + M p k (1 p k ) t k Jeff S. Shamma Asymmetric Information Security Games 24/43

50 Uninformed player: Defend vs guarantee Defend: React to σ s.t. γ t (σ, BR(σ)) l, σ e.g., Belief based reaction Jeff S. Shamma Asymmetric Information Security Games 25/43

51 Uninformed player: Defend vs guarantee Defend: React to σ s.t. γ t (σ, BR(σ)) l, σ e.g., Belief based reaction Guarantee: Find τ s.t. γ t (σ, τ) l, σ Jeff S. Shamma Asymmetric Information Security Games 25/43

52 Uninformed player: Defend vs guarantee Defend: React to σ s.t. γ t (σ, BR(σ)) l, σ e.g., Belief based reaction Guarantee: Find τ s.t. γ t (σ, τ) l, σ Example: L R T 1-1 B -1 1 Jeff S. Shamma Asymmetric Information Security Games 25/43

53 Uninformed player: Defend vs guarantee Defend: React to σ s.t. γ t (σ, BR(σ)) l, σ e.g., Belief based reaction Guarantee: Find τ s.t. γ t (σ, τ) l, σ Example: L R T 1-1 B -1 1 Defend: BR(T) = R & BR(B) = L & BR(50-50) = L/R (l = 0) Jeff S. Shamma Asymmetric Information Security Games 25/43

54 Uninformed player: Defend vs guarantee Defend: React to σ s.t. γ t (σ, BR(σ)) l, σ e.g., Belief based reaction Guarantee: Find τ s.t. γ t (σ, τ) l, σ Example: L R T 1-1 B -1 1 Defend: BR(T) = R & BR(B) = L & BR(50-50) = L/R (l = 0) Guarantee: (l = 0) Jeff S. Shamma Asymmetric Information Security Games 25/43

55 Approachability strategy for Col Hypothetical payoff vector: Given observations Col can compute g t (a t, b t ) = ( M 1 (a t, b t ) M 2 (a t, b t )... M K (a t, b t ) ) as well as its running average g t+1 = g t + 1 t + 1 (g t(a t, b t ) g t ) Jeff S. Shamma Asymmetric Information Security Games 26/43

56 Approachability strategy for Col Hypothetical payoff vector: Given observations Col can compute g t (a t, b t ) = ( M 1 (a t, b t ) M 2 (a t, b t )... M K (a t, b t ) ) as well as its running average g t+1 = g t + 1 t + 1 (g t(a t, b t ) g t ) Challenge: Steer g t so that lim sup p T g t Cavu(p) t Jeff S. Shamma Asymmetric Information Security Games 26/43

57 Blackwell approachability g t+1 = g t + 1 t + 1 (g t(a t, b t ) g t ) Approachability: A closed convex set, C, is approachable, i.e., Pr [dist(g t, C) 0] = 1 if and only if for all half-spaces, H, containing C, there exists a y such that { Co y b g t (a, b) } a A H b * Jeff S. Shamma Asymmetric Information Security Games 27/43

58 Approachability strategy for Col, cont Construction: 1 Find a supporting hyperplane v R K such that v T p = Cavu(p) u(q) v T q, q 2 Define C = { x R K x v } 3 At stage t, if g t C, define q = 1 Z ( ) g t Π(g t, C) and play optimal strategy for D(q) q * g t Jeff S. Shamma Asymmetric Information Security Games 28/43

59 Recap Row belief splitting: Cavu(p) v t (p) Jeff S. Shamma Asymmetric Information Security Games 29/43

60 Recap Row belief splitting: Cavu(p) v t (p) Col naive defense: v t (p) Cavu(p) + M p k (1 p k ) t k Jeff S. Shamma Asymmetric Information Security Games 29/43

61 Recap Row belief splitting: Cavu(p) v t (p) Col naive defense: v t (p) Cavu(p) + M p k (1 p k ) t k Col approachability: lim sup E [γ t (σ, τ)] Cavu(p) t Jeff S. Shamma Asymmetric Information Security Games 29/43

62 Recap Row belief splitting: Cavu(p) v t (p) Col naive defense: v t (p) Cavu(p) + M p k (1 p k ) t k Col approachability: lim sup E [γ t (σ, τ)] Cavu(p) t Implication: Infinite horizon game, Γ (p) has value Cavu(p) Jeff S. Shamma Asymmetric Information Security Games 29/43

63 Computations: Face value Suppose M is S S Assume oblivious Row who ignores Col s actions (optimal) Jeff S. Shamma Asymmetric Information Security Games 30/43

64 Computations: Face value Suppose M is S S Assume oblivious Row who ignores Col s actions (optimal) Number of strategies: Stage 0: state action: S K Stage 1: (state,action) action: S K S. Stage T : (state, action,..., action) action: S K ST Total: T t=0 S K St Conversion to strategic form computationally prohibitive Jeff S. Shamma Asymmetric Information Security Games 30/43

65 Recursive structure ( v t+1 (p) = 1 t + 1 max X min y p k x k M k y + t a k π(a; X, p)v t (B(a; X, p)) ) Row is oblivious to Col Col plays myopic defense based on current beliefs Jeff S. Shamma Asymmetric Information Security Games 31/43

66 Recursive structure intuition What is Col s best response to z t+1 = Z t (z t, a t ) a t X t (z t ) Jeff S. Shamma Asymmetric Information Security Games 32/43

67 Recursive structure intuition What is Col s best response to z t+1 = Z t (z t, a t ) a t X t (z t ) Value iteration: Col plays myopic defense V t (z t, p t ) = min y V T (z T, p T ) = min E [ M k (a T, b T ) ] b T y = min pt k X k (z T ) T M k y y pt k X k (z t ) T M k y k k + E [V t+1 (Z t (z t, a t ), B(a t ; X(z t ), p t ))] Note: Reversed time indexing and neglected normalization. Jeff S. Shamma Asymmetric Information Security Games 32/43

68 Recursive structure intuition, cont V t (z t, p t ) = min y pt k X k (z t ) T M k y + E [V t+1 (Z t (z t, a t ), B(a t ; X(z t ), p t ))] k Jeff S. Shamma Asymmetric Information Security Games 33/43

69 Recursive structure intuition, cont V t (z t, p t ) = min y pt k X k (z t ) T M k y + E [V t+1 (Z t (z t, a t ), B(a t ; X(z t ), p t ))] k Row s task as a maximizer v t (p t ) = max X v T (p T ) = max X k min y pt k x k M k y k min y p k t x k M k y + E [v t+1 (B(a t ; X, p t ))] Jeff S. Shamma Asymmetric Information Security Games 33/43

70 Recursive structure intuition, cont V t (z t, p t ) = min y pt k X k (z t ) T M k y + E [V t+1 (Z t (z t, a t ), B(a t ; X(z t ), p t ))] k Row s task as a maximizer v t (p t ) = max X v T (p T ) = max X Equivalent problem: State space: p t+1 = B(p t, a t) Action space: (A) K Stage reward: min y k x k M k y k min y pt k x k M k y k min y p k t x k M k y + E [v t+1 (B(a t ; X, p t ))] Jeff S. Shamma Asymmetric Information Security Games 33/43

71 LP for v 1 (p) S K + 1 variables & S constraints ( max min p k (x k ) T M k) y X K y k max l X,l p k x k M k l 1 T k x k, k Jeff S. Shamma Asymmetric Information Security Games 34/43

72 LP for v 1 (p) S K + 1 variables & S constraints Extended v 1 ( ): ( max min p k (x k ) T M k) y X K y k max l X,l p k x k M k l 1 T k x k, k Redefine v 1 (q) over positive q R K + Positive homogeneity: c v 1 (q) = v 1 (c q) Jeff S. Shamma Asymmetric Information Security Games 34/43

73 LP for v 2 (p)? v 2 (p) = max X = max X min θ y k min θ y k p k x k M k y + (1 θ) a p k x k M k y + (1 θ) a π(a; X, p)v 1 (B(a; X, p)) v 1 (x 1 a p 1,..., x K a p K ) Jeff S. Shamma Asymmetric Information Security Games 35/43

74 LP for v 2 (p)? v 2 (p) = max X min θ y k p k x k M k y + (1 θ) a π(a; X, p)v 1 (B(a; X, p)) = max X min θ y k p k x k M k y + (1 θ) a v 2 (p) = max θl 0 + (1 θ) X,l 0,...,l A a p k x k M k l 0 1 T k v 1 (x 1 a p 1,..., x K a p K ) l a, a x k, k v 1 (x 1 a p 1,..., x K a p K ) l a Jeff S. Shamma Asymmetric Information Security Games 35/43

75 LP for v 2 (p)? v 2 (p) = max X min θ y k p k x k M k y + (1 θ) a π(a; X, p)v 1 (B(a; X, p)) = max X min θ y k p k x k M k y + (1 θ) a v 2 (p) = max θl 0 + (1 θ) X,l 0,...,l A a p k x k M k l 0 1 T k v 1 (x 1 a p 1,..., x K a p K ) l a, a x k, k v 1 (x 1 a p 1,..., x K a p K ) l a Constraints on v 1 ( ) result in product terms in LP Jeff S. Shamma Asymmetric Information Security Games 35/43

76 LHS vs RHS LHS Nested LP: max c T v v Av b v i F i w f i Jeff S. Shamma Asymmetric Information Security Games 36/43

77 LHS vs RHS LHS Nested LP: max c T v v Av b v i F i w f i LP structure lost! Jeff S. Shamma Asymmetric Information Security Games 36/43

78 LHS vs RHS LHS Nested LP: max c T v v Av b v i F i w f i LP structure lost! RHS Nested LP: max c T v v Av b Fw i v i f i LP structure preserved! Jeff S. Shamma Asymmetric Information Security Games 36/43

79 LP for v 1 (p) revisited v 1 (p) = max l X,l p k x k M k l 1 T k x k, k v 1 (p) = max l Z,l z k M k l 1 T k 1 T z k = p k, k Change of variables: z k = p k x k 0 Probabilities now enter in RHS Jeff S. Shamma Asymmetric Information Security Games 37/43

80 LP for v 2 (p) revisited v 2 (p) = max θl 0 + (1 θ) X,l 0,...,l A a p k x k M k l 0 1 T k v 1 (x 1 a p 1,..., x K a p K ) l a, a x k, k Each constraint on v 1 ( ) is an LP: z k (a)m k l a 1 T k 1 T z k (a) = x k a p k l a (S + 1) (S K vars & S cons ) Jeff S. Shamma Asymmetric Information Security Games 38/43

81 LP computations Claim: Recursive structure: If v t(p) has RHS LP-dependence on p, then v t+1(p) has RHS LP-dependence on p. Growth in LP size: LP size grows with size[v t] S size[v t 1] Jeff S. Shamma Asymmetric Information Security Games 39/43

82 LP computations Claim: Recursive structure: If v t(p) has RHS LP-dependence on p, then v t+1(p) has RHS LP-dependence on p. Growth in LP size: LP size grows with size[v t] S size[v t 1] Polynomial dependence on size of game (but not length): (S + S S T ) K vs T t=0 S K St }{{} face value (cf., sequence form of Koller, von Stengel, & Megiddo, 1996) Jeff S. Shamma Asymmetric Information Security Games 39/43

83 LP computations Claim: Recursive structure: If v t(p) has RHS LP-dependence on p, then v t+1(p) has RHS LP-dependence on p. Growth in LP size: LP size grows with size[v t] S size[v t 1] Polynomial dependence on size of game (but not length): (S + S S T ) K vs T t=0 S K St }{{} face value (cf., sequence form of Koller, von Stengel, & Megiddo, 1996) Recursive computation applicable to time-varying repeated games (i.e., changing M-matrices) Jeff S. Shamma Asymmetric Information Security Games 39/43

84 Illustration: Example I L R T 1 0 B 0 0 M 1 L R T 0 0 B 0 1 M Jeff S. Shamma Asymmetric Information Security Games 40/43

85 Illustration: Network interdiction System: One high capacity resource and several low capacity (unknown to attacker) Attacker: Observes usage (binary) during Phase I Disables selected resource for Phase II Tension: Initial vs future usage Note: Time-varying M-matrices Jeff S. Shamma Asymmetric Information Security Games 41/43

86 Network interdiction, cont There is one high capacity channel, and the attacker can block one channel from stage 4 to 11 2 value of the game total number of channels Initial phase: 3 stages Remaining phase: R stages Probability of using high capacity resource: r \ t Jeff S. Shamma Asymmetric Information Security Games 42/43

87 Concluding remarks Recap: Examples Basic results Computational approach Jeff S. Shamma Asymmetric Information Security Games 43/43

88 Concluding remarks Recap: Examples Basic results Computational approach Extensions: Discounted problems: γ λ (σ, τ) = E [ (1 λ) ] λ t M k (a t, b t) t=0 Markov chains with informed controller: Receding horizon implementation k t+1 φ(k t, a t) Jeff S. Shamma Asymmetric Information Security Games 43/43

89 Concluding remarks Recap: Examples Basic results Computational approach Extensions: Discounted problems: γ λ (σ, τ) = E [ (1 λ) ] λ t M k (a t, b t) t=0 Markov chains with informed controller: Receding horizon implementation k t+1 φ(k t, a t) Lingering issue: Computational policies for uninformed player Jeff S. Shamma Asymmetric Information Security Games 43/43